Framework

XOR Transformations

How agents traverse the hypercube through codeword-driven state transitions

XOR Transformation Flow

The Transformation Mechanism

Agents move through the 512-vertex state space by XOR with carefully chosen codewords. Given state x and codeword c:

x′ = x ⊕ c

This flips specific bits according to the codeword pattern, moving the agent to a new vertex.

Doubly-Even Codewords

Each codeword has Hamming weight divisible by 4, preserving algebraic structure inherited from supersymmetric adinkra graphs.

c₁ = [1,1,1,1,0,0,0,0,0] weight 4 c₂ = [1,1,0,0,1,1,0,0,0] weight 4 c₃ = [1,0,1,0,1,0,1,0,0] weight 4 c₄ = [1,0,0,1,1,0,0,1,0] weight 4 c₅ = [1,1,1,1,1,1,1,1,0] weight 8 c₆ = [0,0,0,0,1,1,1,1,0] weight 4

The Averaging Operator

The averaging operator T is idempotent (T² = T) — a projection onto C-invariant functions:

T f(x) = (1/|C|) ∑c ∈ C f(x ⊕ c)   |   T² = T

Since C is a linear code (closed under XOR), for any fixed c, as d ranges over C, {c ⊕ d} = C. This makes T² = T.

Key Properties

  • IdempotentApplying twice gives the same result as once
  • LinearT(af + bg) = aT(f) + bT(g)
  • ProjectionProjects onto C-invariant functions — patterns stable under all transformations